OFFSET
0,2
COMMENTS
Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 9.
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (7,42,6,-81,27).
FORMULA
G.f.: (1 - 5*x - 11*x^2 + 7*x^3) / (1 - 7*x - 42*x^2 - 6*x^3 + 81*x^4 - 27*x^5).
Recurrence 1:
a(n) = 2*a(n-1) + b(n-1) + c(n-1) + 13*a(n-2) + 2*b(n-2) + c(n-2) + 2*d(n-2),
b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1),
c(n) = 16*a(n-1) + 6*b(n-1) + c(n-1) + 2*e(n-1),
d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1),
e(n) = 16*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1),
with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.
Recurrence 2:
a(n) = 7*a(n-1) + 42*a(n-2) + 6*a(n-3) - 81*a(n-4) + 27*a(n-5) for n >= 5.
For n < 5, recurrence 1 can be used.
EXAMPLE
a(2)=45
1) Two parallel trominos and one domino: There are 3 middle axes of the 2 X 2 cube with 4 rotation images each: 12 images.
___ ___ ___ ___
/__ /| / /| /__ / /|
/__ /| |___ /__ / | /__ /__ / |
| | |/__ /| | | / | | | /|
| |/__ /| | + |___|/ = | |___|/| |
| | |/ | | |/
|_______|/ |_______|/
2) Two "linked" trominos and one domino: 12 rotation images and, as there is no symmetry plane, 12 mirror images: 24 images.
___ ___ ___ ___
/ /| / /| / / /|
/__ / | _______ /__ / | /__ /__ / |
| | / /__ /| | | / | | | /|
| | | + | /__ / | + |___|/ = | |___|/ |
| | | |_| | / | | | /
|___|/ |___|/ |___|___|/
3) Using only dominos: A006253(2)=9 ways, Sum: a(2) = 12 + 24 + 9 = 45.
MATHEMATICA
LinearRecurrence[{7, 42, 6, -81, 27}, {1, 2, 45, 412, 4705}, 25] (* Paolo Xausa, Oct 02 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gerhard Kirchner, Jan 30 2023
STATUS
approved